Solving the Differential Equation: (d^3 + 2d^2 + d)y = e^(2x) + x^2 + x + sin(2x)
This article will guide you through solving the given thirdorder linear nonhomogeneous differential equation:
(d^3 + 2d^2 + d)y = e^(2x) + x^2 + x + sin(2x)
We will break down the solution process into distinct steps.
1. Finding the Complementary Solution (y_c)

Step 1: Find the roots of the auxiliary equation:
The auxiliary equation is obtained by replacing the differential operators (d) with the variable (m) in the homogeneous part of the equation:
m^3 + 2m^2 + m = 0
Factoring out 'm' gives: m(m^2 + 2m + 1) = 0
Further factoring: m(m + 1)^2 = 0
This gives us roots: m = 0, m = 1 (repeated root)

Step 2: Form the complementary solution (y_c)
For distinct roots, we use the term 'e^(mx)' in the solution. For repeated roots, we multiply the term by 'x'.
Therefore, the complementary solution is: y_c = c_1 + c_2e^(x) + c_3xe^(x)
2. Finding the Particular Solution (y_p)

Step 1: Identify the form of the particular solution based on the nonhomogeneous terms:
The nonhomogeneous terms are: e^(2x) + x^2 + x + sin(2x)
We need to consider each term separately:
 e^(2x): Since '2' is not a root of the auxiliary equation, we assume a particular solution of the form Ae^(2x).
 x^2 + x: Since the auxiliary equation has a root of '0', we assume a particular solution of the form Bx^2 + Cx + D.
 sin(2x): We assume a particular solution of the form E sin(2x) + F cos(2x).

Step 2: Combine the individual forms and adjust for repetition:
The general form of the particular solution is:
y_p = Ae^(2x) + Bx^2 + Cx + D + E sin(2x) + F cos(2x)

Step 3: Determine the coefficients (A, B, C, D, E, F) by substituting y_p into the original equation:
Substitute y_p and its derivatives into the equation (d^3 + 2d^2 + d)y = e^(2x) + x^2 + x + sin(2x) and solve for the constants. This involves differentiation, substitution, and equating coefficients.
3. Combining the Solutions
The general solution (y) is the sum of the complementary solution (y_c) and the particular solution (y_p):
y = y_c + y_p
y = c_1 + c_2e^(x) + c_3xe^(x) + Ae^(2x) + Bx^2 + Cx + D + E sin(2x) + F cos(2x)
Conclusion
This process provides the general solution to the given differential equation. Remember that the constants (c_1, c_2, c_3, A, B, C, D, E, F) need to be determined using initial conditions or boundary values, if provided.